Lecture 4 (Pdf)

GFD 2006 Lecture 4: Interfacial instability in two-component melts Grae Worster; notes by Shane Keating and Takahide Okabe March 15, 2007 So far, we have looked at some of the fundamentals associated with solidification of pure melts. When we try to solidify a solution of two or more components, salt and water, for example, the character of the solidification changes considerably. In particular, the presence of salt can depress the temperature at which ice and salt water can coexist in thermal equilibrium. This has an important consequence for the growth of sea ice: unless there is some other mechanism for the transport of the salt field, such as convection, the growth of the ice is limited by the rate at which excess salt can diffuse away from the interface. Finally, we will discuss the morphological instability in two-component melts. We shall see that the solute field is destabilizing and can give rise to morphological instability even when the liquid phase is not initially supercooled. 1 Two-component melts 1.1 A simple demonstration We shall begin with a simple demonstration. Crushed ice at 0◦C is placed in a cup with a thermometer. We add a handful of salt at room temperature and stir briskly. The ice begins to melt, but what happens to the temperature? We notice that there is some melt water in the cup, which helps bring the ice and salt into contact, and see a fairly rapid decrease in the temperature measured by the thermometer: after a few minutes, it reads almost 10 C. What's happening here is not melting. Rather, − ◦ we are observing dissolution of the pure ice into the mixture of salt and water. In this lecture, we will attempt to make more explicit the distinction between melting and dissolution. 1.2 Equilibrium phase diagrams In Figure 1, we show the equilibrium phase diagram for a simple 2-component mixture, or binary melt { in this case, salt and water. The equilibrium state of a given mixture of salt and water at temperature T and composition C (i.e., concentration of salt) and at constant pressure can be represented on this diagram by the point (T; C). The phase diagram is divided into regions of different phase; this diagram is \simple" in the sense that there are only two possible solid phases: pure ice, or solid salt. In Figure 1 these lie along the vertical axes at 0% and 100% concentration respectively. Apart from these two solid phases, we can 31 Figure 1: Equilibrium phase diagram for a solution of salt and water. also form a liquid solution of the two end members (i.e., salt or water), or some liquid/solid mixture of the two substances. Other materials have more complicated solid phases and neccessarily more complicated phase diagrams, which we will examine briefly later. The curved line in Figure 1 is the liquidus, representing the temperature at which a binary melt of a given composition C can exist in equilibrium in both the liquid and solid phase. For 0% salt concentration, the liquidus temperature is simply the melting point of ice { 0◦C { while for 100% salt it is 801◦C. When we contaminate pure water at 0◦C with a small amount of salt, the equilibrium freezing temperature is lowered. Thus, when we added a small amount of salt to the ice in our experiment, we saw that we still had liquid even at temperatures as low as 10 C. − ◦ Equivalently, one could start with pure molten salt at 801◦C and contaminate it with a small amount of water to lower the melting point. The two liquidus curves meet at a point (TE; CE), called the eutectic: this is the minimum temperature at which solid and liquid saltwater can coexist in thermodynamic equilibrium1. If we slowly change the temperature or composition, the mixture will trace a trajectory on the phase diagram, as shown in Figure 1 for the case of seawater. We start by cooling seawater to 2 C where it reaches the liquidus curve T (C) and starts to freeze. Below − ◦ L this temperature, we start to form pure solid ice in equilibrium with seawater of higher concentration. As more and more solid ice is formed, less water is available and so the salt concentration increases steadily. We can invert the liquidus curve T = TL (C) to find the 1According to one popular story, German physicist Gabriel Fahrenheit (1686-1736) chose the triple eutectic temperature of water, salt and ammonium chloride, being the lowest temperature he could achieve in his laboratory, as the zero of his eponymous scale. Both Fahrenheit and Celsius are centrigrade scales: An- ders Celsius (1701-1744) chose 100◦C to correspond with the boiling point of water at sea level; Fahrenheit ◦ likewise chose a reliable, easily reproducible, steady temperature for 100 F { the anal temperature of his horse. It should be noted, however, that wikipedia.org lists no less than six competing versions of the same story, so at the risk of punning, one should be advised to take such apocryphal tales with a pinch of salt. 32 Figure 2: A more generic phase diagram. See text for details. composition of the remaining liquid: C = CL (T ). It is worthwhile to extend the simple phase diagram for salt and water to one more typical of other binary melts, as shown in Figure 2. In addition to the liquidus, there is a solidus further subdividing the phase diagram. There are now four distinct phases, which we describe below. Region I is a liquid solution of the two end members. In region II, the mixture is in a solid solution, where the end members are mixed on the lattice scale. An example of this is the silicate compound olivine, (Fe;Mg)2 SiO4, although the phase diagram is quite different from the one shown in Figure 2. Iron and magnesium sit fairly equally in the lattice sites and will occur in different proportions depending upon the temperature. In contrast, salt and water do not form a solid solution, and will exist in the solid phase only as pure substances, at least as far as we are concerned in this course. In region III, the solid solution and the liquid phase coexist in equilibrium. Finally, in region IV, we have a mixture of crystals of the two end members: i.e., pure ice coexisting with pure salt crystals. In addition, there are regions of the equilibrium phase diagram mirroring region III, where pure crystals of one end members coexist with a solid solution of both end members. The exact location in the equilibrium phase diagram of the transition to this region, indicated in Figure 2 by a dashed line, is difficult to measure experimentally, because the compositional relaxation times below the eutectic are on the order of geological timescales. We ignore such detailed structure in our analysis. Thus, the equilibrium phase diagram can tell us a great deal about what proportion of a mixture is in what phase, and what can coexist in equilibrium. However, it cannot tell us anything about the geometry of the solid phase formed; whether the ice forms in layers, or a slurry of ice crystals and salt water, or in the form of a mushy layer of dendritic ice crystals separated by interstitial seawater, as we shall examine in the next lecture. The microscopic details of the distribution of the phases depends strongly on how you lower the 33 temperature; however, the ratio of the phases will not depend on the history of the mixture. 1.3 A few approximations Before we conclude this section, let us introduce some terminology and a few approximations. Firstly, we shall assume (when necessary) that the liquidus can be approximated by a straight line T T mC (1) L ≈ m − and that the solidus concentration is C (T ) k C (T ) (2) S ≈ D L The parameter kD is called the distribution coefficient, and is approximately zero for a salt and water solution. Thus we will assume that the solution will form only pure cystals of salt or ice. 2 Solidification of sea ice 2.1 The Stefan problem for a salt water solution Let us now revisit the Stefan problem2; this time, however, we consider the case of salt water in contact with a boundary at a temperature below the liquidus temperature of the solution, as depicted in Figure 3 We denote by Ti and Ci respectively the interfacial temperature and composition of the salt water, to be determined. We further demand that the ice and the salt water at the interface are in thermodynamic equilibrium so that Ti is the liquidus temperature and Ti = TL (Ci) : (3) This is in contrast to the Stefan problem where the interfacial temperature was simply the melting temperature of pure ice Tm. Here, however, the temperature at which the salt water freezes is set by the interfacial concentration of salt, and we shall see that the rate at which the interface advances is limited by the rate at which we can remove excess salt from the region near the interface. The composition of salt inside the ice will be zero, as discussed above; however, we shall denote it by CS to be a little more general. The far field temperature and composition of the sea water are T and C0 respectively. The boundary temperature TB will be below the 1 liquidus temperature of the undisturbed solute field: TB < TL (C0). The equations to be solved are the diffusion equation in the ice and the sea water @T @2T = κ in x < a and x > a (4) @t @x2 2Note that in this treatment, we will neglect the effects of both kinetics and surface energy.

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